Aspherical pro-$p$-groups

O. V. Mel'nikov

Sbornik. Mathematics, Tome 193 (2002) no. 11, pp. 1639-1670

Résumé

The notion of an aspherical pro-$p$-group is introduced. It is proved that if a group $G=F/N$ is aspherical, where $F$ is a free pro-$p$-group, then the relation $\mathbb F_p[[G]]$-module $\overline N=N/N^p[N,N]$ satisfies an assertion of the type of Lyndon's identity theorem. The finite subgroups and the centre of $G$ are described. The structure of an aspherical pro-$p$-group $G$ with a soluble normal subgroup $A\ne\{1\}$ is studied. In particular, if $A\cong\mathbb Z_p$, then $G$ contains a subgroup of finite index of the form $A\leftthreetimes W$ where $W$ is a free pro-$p$-group.

Bibliographie

[1] Lindon R., Shupp P., Kombinatornaya teoriya grupp, Mir, M., 1980 | MR

[2] Huebschmann J., “Cohomology theory of aspherical groups and of small cancellation groups”, J. Pure Appl. Algebra, 14:2 (1979), 137–143 | DOI | MR | Zbl

[3] Chiswell I. M., Collins D. J., Huebschmann J., “Aspherical group presentations”, Math. Z., 178:1 (1981), 1–36 | DOI | MR | Zbl

[4] Kokh Kh., Teoriya Galua $p$ -rasshirenii, Mir, M., 1973 | MR

[5] Melnikov O. V., “Podgruppy i gomologiii svobodnykh proizvedenii prokonechnykh grupp”, Izv. AN SSSR. Ser. matem., 53:1 (1989), 97–120 | MR | Zbl

[6] Gildenhuys D., Ribes L., “Profinite groups and Boolean graphs”, J. Pure Appl. Algebra, 12:1 (1978), 21–47 | DOI | MR | Zbl

[7] Zalesskii P. A., Melnikov O. V., “Podgruppy prokonechnykh grupp, deistvuyuschikh na derevyakh”, Matem. sb., 135:4 (1988), 419–439

[8] Zalesskii P. A., Melnikov O. V., “Fundamentalnye gruppy grafov prokonechnykh grupp”, Algebra i analiz, 1:4 (1989), 117–135 | MR

[9] Brumer A., “Pseudo-compact algebras, profinite groups and class formations”, J. Algebra, 4:3 (1966), 442–470 | DOI | MR | Zbl

[10] Melnikov O. V., “Faktorgruppy prokonechnykh grupp s dvoistvennostyu Puankare”, Izv. AN Belarusi. Ser. fiz.-matem. nauk, 1996, no. 3, 54–58 | Zbl

[11] Gruenberg K. W., “Projective profinite groups”, J. London Math. Soc., 42:1 (1967), 155–165 | DOI | MR | Zbl

[12] Gildenhuys D., Lim C.-K., “Free pro -$\mathscr C$-groups”, Math. Z., 125:1 (1972), 233–254 | DOI | MR | Zbl

[13] Fox R. H., “Free differential calculus. I: Derivation in the free group ring”, Ann. of Math. (2), 57:3 (1953), 547–560 | DOI | MR | Zbl

[14] Gruenberg K. W., Cohomological topics in group theory, Lecture Notes in Math., 143, Springer-Verlag, Berlin, 1970 | MR

[15] Bass Kh., Algebraicheskaya $K$-teoriya, Mir, M., 1973 | MR | Zbl

[16] Serre J.-P., “Cohomologie des groupes discrets”, Ann. of Math. Stud., 70 (1971), 77–169 | MR | Zbl

[17] Burbaki N., Algebra, Gl. X. Gomologicheskaya algebra, Nauka, M., 1987 | MR

[18] Ribes L., “On amalgamated products of profinite groups”, Math. Z., 123:4 (1971), 357–374 | DOI | MR

[19] Binz E. N., Neukirch J., Wenzel G. H., “A subgroup theorem for free products of pro-finite groups”, J. Algebra, 19:1 (1971), 104–109 | DOI | MR | Zbl

[20] Haran D., “On closed subgroups of free products of profinite groups”, Proc. London Math. Soc., 123:2 (1987), 266–298 | MR

[21] Melnikov O. V., “O svobodnykh proizvedeniyakh absolyutnykh grupp Galua”, Sib. matem. zhurn., 40:1 (1999), 113–118 | MR | Zbl

[22] Serr Zh.-P., Kogomologii Galua, Mir, M., 1968 | MR

[23] Labute J. P., “Classification of Demushkin groups”, Canad. J. Math., 19:1 (1967), 106–132 | MR | Zbl

[24] Melnikov O. V., “Normalnye deliteli svobodnykh prokonechnykh grupp”, Izv. AN SSSR. Ser. matem., 42:1 (1978), 3–25 | MR | Zbl

[25] Melnikov O. V., Shishkevich A. A., “Pro -$p$ -gruppy s virtualnoi dvoistvennostyu Puankare razmernosti $2$”, Dokl. NAN Belarusi, 46:1 (2002), 13–15 | MR